Started progress on the multidimensional kalman filter

Kalman Filters.ipynb

@@ -50,7 +50,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 34
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -74,11 +74,11 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "0.0883 1.1333 1.3527 -0.1540 1.0787 1.1940 -0.4908 0.9334 -1.3604 -1.5230 -1.1513 -1.0829 0.5573 0.7816 0.0587 -0.0605 0.0651 -0.4040 0.3429 0.2245\n"
+        "0.9827 0.4385 1.6524 0.6318 -0.2302 -1.0772 -0.0953 0.1147 1.1092 1.6671 0.1331 0.5139 -0.5076 -0.9376 -0.7724 0.4346 -0.3805 -1.9624 -0.3980 -0.9907\n"
        ]
       }
      ],
-     "prompt_number": 35
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -117,7 +117,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2626cd0>"
+        "<matplotlib.figure.Figure at 0x28aa590>"
        ]
       },
       {
@@ -128,7 +128,7 @@
        ]
       }
      ],
-     "prompt_number": 36
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -167,13 +167,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2d34b90>"
+        "<matplotlib.figure.Figure at 0x2e8cb50>"
        ]
       }
      ],
-     "prompt_number": 37
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -200,13 +200,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2a0f350>"
+        "<matplotlib.figure.Figure at 0x28b11d0>"
        ]
       }
      ],
-     "prompt_number": 38
+     "prompt_number": 16
     },
     {
      "cell_type": "code",
@@ -220,13 +220,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAAEZCAYAAACJjGL9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2ebbf50>"
+        "<matplotlib.figure.Figure at 0x2917e10>"
        ]
       }
      ],
-     "prompt_number": 39
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -259,11 +259,11 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAEACAYAAABS29YJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x23a25d0>"
+        "<matplotlib.figure.Figure at 0x2d9ff90>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -281,10 +281,10 @@
      "collapsed": false,
      "input": [
       "dog = dog_sensor (23, 0, 5)\n",
-      "ys = range (100)\n",
-      "xs = []\n",
-      "for i in ys:\n",
-      "    xs.append (dog.sense())\n",
+      "xs = range (100)\n",
+      "ys = []\n",
+      "for i in xs:\n",
+      "    ys.append (dog.sense())\n",
       "               \n",
       "plot (xs,ys)\n",
       "show()"
@@ -295,13 +295,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2a20490>"
+        "<matplotlib.figure.Figure at 0x2e5a390>"
        ]
       }
      ],
-     "prompt_number": 41
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -379,11 +379,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2edc850>"
+        "<matplotlib.figure.Figure at 0x2e8f250>"
        ]
       }
      ],
-     "prompt_number": 42
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -424,11 +424,11 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAEACAYAAABS29YJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2d31810>"
+        "<matplotlib.figure.Figure at 0x2e6d650>"
        ]
       }
      ],
-     "prompt_number": 56
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -463,7 +463,8 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -481,7 +482,8 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -511,30 +513,30 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "time: 0 position =  1.132 variance =  2.500\n",
-        "time: 1 position =  0.798 variance =  1.667\n",
-        "time: 2 position =  0.279 variance =  1.250\n",
-        "time: 3 position =  0.121 variance =  1.000\n",
-        "time: 4 position =  0.017 variance =  0.833\n",
-        "time: 5 position =  0.071 variance =  0.714\n",
-        "time: 6 position =  0.213 variance =  0.625\n",
-        "time: 7 position =  0.191 variance =  0.556\n",
-        "time: 8 position =  0.223 variance =  0.500\n",
-        "time: 9 position =  0.257 variance =  0.455\n",
-        "time: 10 position =  0.398 variance =  0.417\n",
-        "time: 11 position =  0.450 variance =  0.385\n",
-        "time: 12 position =  0.537 variance =  0.357\n",
-        "time: 13 position =  0.603 variance =  0.333\n",
-        "time: 14 position =  0.548 variance =  0.312\n",
-        "time: 15 position =  0.624 variance =  0.294\n",
-        "time: 16 position =  0.566 variance =  0.278\n",
-        "time: 17 position =  0.492 variance =  0.263\n",
-        "time: 18 position =  0.505 variance =  0.250\n",
-        "time: 19 position =  0.568 variance =  0.238\n"
+        "time: 0 position =  1.497 variance =  2.500\n",
+        "time: 1 position =  0.809 variance =  1.667\n",
+        "time: 2 position =  0.817 variance =  1.250\n",
+        "time: 3 position =  0.485 variance =  1.000\n",
+        "time: 4 position =  0.576 variance =  0.833\n",
+        "time: 5 position =  0.681 variance =  0.714\n",
+        "time: 6 position =  0.650 variance =  0.625\n",
+        "time: 7 position =  0.618 variance =  0.556\n",
+        "time: 8 position =  0.763 variance =  0.500\n",
+        "time: 9 position =  0.728 variance =  0.455\n",
+        "time: 10 position =  0.611 variance =  0.417\n",
+        "time: 11 position =  0.546 variance =  0.385\n",
+        "time: 12 position =  0.529 variance =  0.357\n",
+        "time: 13 position =  0.541 variance =  0.333\n",
+        "time: 14 position =  0.527 variance =  0.312\n",
+        "time: 15 position =  0.432 variance =  0.294\n",
+        "time: 16 position =  0.351 variance =  0.278\n",
+        "time: 17 position =  0.326 variance =  0.263\n",
+        "time: 18 position =  0.337 variance =  0.250\n",
+        "time: 19 position =  0.350 variance =  0.238\n"
        ]
       }
      ],
-     "prompt_number": 55
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -593,7 +595,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 58
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -643,47 +645,47 @@
        "stream": "stdout",
        "text": [
         "UPDATE: 1.0000 , 502.0000\n",
-        "SENSE: 1.9162 , 9.8047\n",
+        "SENSE: 2.5641 , 9.8047\n",
         "\n",
-        "UPDATE: 2.9162 , 11.8047\n",
-        "SENSE: 3.8256 , 5.4138\n",
+        "UPDATE: 3.5641 , 11.8047\n",
+        "SENSE: 0.8489 , 5.4138\n",
         "\n",
-        "UPDATE: 4.8256 , 7.4138\n",
-        "SENSE: -0.0667 , 4.2574\n",
+        "UPDATE: 1.8489 , 7.4138\n",
+        "SENSE: 2.1796 , 4.2574\n",
         "\n",
-        "UPDATE: 0.9333 , 6.2574\n",
-        "SENSE: 5.1593 , 3.8490\n",
+        "UPDATE: 3.1796 , 6.2574\n",
+        "SENSE: 3.1866 , 3.8490\n",
         "\n",
-        "UPDATE: 6.1593 , 5.8490\n",
-        "SENSE: 4.9393 , 3.6904\n",
+        "UPDATE: 4.1866 , 5.8490\n",
+        "SENSE: 3.7578 , 3.6904\n",
         "\n",
-        "UPDATE: 5.9393 , 5.6904\n",
-        "SENSE: 5.9086 , 3.6267\n",
+        "UPDATE: 4.7578 , 5.6904\n",
+        "SENSE: 4.2545 , 3.6267\n",
         "\n",
-        "UPDATE: 6.9086 , 5.6267\n",
-        "SENSE: 6.2753 , 3.6007\n",
+        "UPDATE: 5.2545 , 5.6267\n",
+        "SENSE: 5.4039 , 3.6007\n",
         "\n",
-        "UPDATE: 7.2753 , 5.6007\n",
-        "SENSE: 8.0741 , 3.5900\n",
+        "UPDATE: 6.4039 , 5.6007\n",
+        "SENSE: 7.8103 , 3.5900\n",
         "\n",
-        "UPDATE: 9.0741 , 5.5900\n",
-        "SENSE: 8.0308 , 3.5856\n",
+        "UPDATE: 8.8103 , 5.5900\n",
+        "SENSE: 8.9480 , 3.5856\n",
         "\n",
-        "UPDATE: 9.0308 , 5.5856\n",
-        "SENSE: 9.3586 , 3.5838\n",
+        "UPDATE: 9.9480 , 5.5856\n",
+        "SENSE: 11.1128 , 3.5838\n",
         "\n"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2d4add0>"
+        "<matplotlib.figure.Figure at 0x2917fd0>"
        ]
       }
      ],
-     "prompt_number": 120
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",
@@ -766,6 +768,7 @@
      "collapsed": false,
      "input": [
       "sensor_error = 30000\n",
+      "movement_error = 2\n",
       "pos = (0,500)\n",
       "\n",
       "dog = dog_sensor (pos[0], velocity=movement, noise=sensor_error)\n",
@@ -773,7 +776,7 @@
       "zs = []\n",
       "ps = []\n",
       "\n",
-      "for i in range(100):\n",
+      "for i in range(1000):\n",
       "    pos = update (pos[0], pos[1], movement, movement_error)\n",
       "    \n",
       "    Z = dog.sense()\n",
@@ -794,19 +797,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x34f2fd0>"
+        "<matplotlib.figure.Figure at 0x28b19d0>"
        ]
       }
      ],
-     "prompt_number": 121
+     "prompt_number": 27
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "In this example the noise is extreme Yet the filter still outputs a nearly straight line! This is an astonishing result! What do you think might be the cause of this performance? If you are not sure, don't worry, we will discuss it latter.\n",
+      "In this example the noise is extreme yet the filter still outputs a nearly straight line! This is an astonishing result! What do you think might be the cause of this performance? If you are not sure, don't worry, we will discuss it latter.\n",
       "\n",
       "Now let's lets look at the results when we make a bad initial estimate of position. To avoid obscuring the results I'll reduce the sensor variance to 30, but set the initial position to 1000m. Can the filter recover from a 1000m initial error?"
      ]
@@ -816,6 +819,7 @@
      "collapsed": false,
      "input": [
       "sensor_error = 30\n",
+      "movement_error = 2\n",
       "pos = (1000,500)\n",
       "\n",
       "dog = dog_sensor (0, velocity=movement, noise=sensor_error)\n",
@@ -844,13 +848,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x286de90>"
+        "<matplotlib.figure.Figure at 0x2e60150>"
        ]
       }
      ],
-     "prompt_number": 133
+     "prompt_number": 28
     },
     {
      "cell_type": "markdown",
@@ -866,6 +870,7 @@
      "collapsed": false,
      "input": [
       "sensor_error = 30000\n",
+      "movement_error = 2\n",
       "pos = (1000,500)\n",
       "\n",
       "dog = dog_sensor (0, velocity=movement, noise=sensor_error)\n",
@@ -894,27 +899,151 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAYYAAAEACAYAAAC3adEgAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x286dad0>"
+        "<matplotlib.figure.Figure at 0x2d9aa10>"
        ]
       }
      ],
-     "prompt_number": 136
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "This time the filter does struggle. Notice that the previous example only computed 100 updates, whereas this example uses 1000. By my eye it takes the filter 400 or so iterations to become reasonable accurate, but maybe over 600 before the results are good.  "
+      "This time the filter does struggle. Notice that the previous example only computed 100 updates, whereas this example uses 1000. By my eye it takes the filter 400 or so iterations to become reasonable accurate, but maybe over 600 before the results are good. Kalman filters are good, but we cannot expect miracles. If we have extremely noisy data and extremely bad initial conditions, this is as good as it gets.\n",
+      "\n",
+      "Finally, let's make the suggest change of making our initial position guess just be the first sensor measurement."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sensor_error = 30000\n",
+      "movement_error = 2\n",
+      "pos = None\n",
+      "\n",
+      "dog = dog_sensor (0, velocity=movement, noise=sensor_error)\n",
+      "\n",
+      "zs = []\n",
+      "ps = []\n",
+      "\n",
+      "for i in range(1000):\n",
+      "    Z = dog.sense()\n",
+      "    zs.append(Z)\n",
+      "    if pos == None:\n",
+      "        pos = (Z, 500)\n",
+      "    \n",
+      "    pos = sense (pos[0], pos[1], Z, sensor_error)\n",
+      "    ps.append(pos[0])\n",
+      "\n",
+      "    pos = update (pos[0], pos[1], movement, movement_error)\n",
+      "\n",
+      "p1, = plot (zs,c='r', linestyle='dashed')\n",
+      "p2, = plot (ps, c='b')\n",
+      "legend ([p1,p2], ['measurement', 'filter'], 2)\n",
+      "show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x2930650>"
+       ]
+      }
+     ],
+     "prompt_number": 30
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This simple change significantly improves the results. On some runs it takes 200 iterations or so to settle to a good solution, but other runs it converges very rapidly. This all depends on whether the initial measurement $Z$ had a small amount or large amount of noise. \n",
+      "\n",
+      "200 iterations may seem like a lot, but the amount of noise we are injecting is truly huge. In the real world we use sensors like thermometers,  laser rangefinders, GPS satellites, computer vision, and so on. None have the enormous error as shown here. A reasonable value for the variance for a cheap thermometer might be 10, for example, and our code is using 30,000 for the variance. "
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "#Explaining the Results"
+      "####Explaining the Results\n",
+      "\n",
+      "So how does the Kalman filter do so well? I have glossed over one aspect of the filter as it becomes confusing to address too many points at the same time. In these example we do not have 1 sensor but 2. The first sensor is the RFID sensor that outputs the position measurement, and the second sensor measures our dog's movement using an intertial tracker. How does our filter perform if that tracker is also noisy? Let's see:\n",
+      "\n"
      ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sensor_error = 30000\n",
+      "movement_sensor = 30000\n",
+      "pos = (0,500)\n",
+      "\n",
+      "dog = dog_sensor (0, velocity=movement, noise=sensor_error)\n",
+      "\n",
+      "zs = []\n",
+      "ps = []\n",
+      "\n",
+      "for i in range(1000):\n",
+      "    Z = dog.sense()\n",
+      "    zs.append(Z)\n",
+      "    \n",
+      "    pos = sense (pos[0], pos[1], Z, sensor_error)\n",
+      "    ps.append(pos[0])\n",
+      "\n",
+      "    pos = update (pos[0], pos[1], movement, movement_error)\n",
+      "\n",
+      "p1, = plot (zs,c='r', linestyle='dashed')\n",
+      "p2, = plot (ps, c='b')\n",
+      "legend ([p1,p2], ['measurement', 'filter'], 2)\n",
+      "show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAYYAAAEACAYAAAC3adEgAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x28aaad0>"
+       ]
+      }
+     ],
+     "prompt_number": 31
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This result is worse than the example where only the measurement sensor was noisy. Instead of being mostly straight, this time the filter's output is distintly jagged. But, it still mostly tracks the dog. What is happening here?\n",
+      "\n",
+      "This illustrates the effects of *multi-sensor fusion*. Suppose the dog is actually at 10.0, and we get subsequent measurement readings of -289.78 and 301.43.  From that information alone it is impossible to tell if the dog is standing still during very noisy measurements, or perhaps sprinting from -289 to 301 and being accurately measured. But we have a second source of information, his velocity. Even when the velocity is also noisy, it constrains what our beliefs might be. For example, suppose with the readings of -289.78 to 301.43 we get a velocity reading of 590. That matches the difference between the two positions quite well, so this will lead us to believe the RFID sensor and the velocity sensor. Now suppose we got a velocity reading of 1.7. This doesn't match our RFID reading very well. Finally, suppose the velocity reading was -678.8. This completely contradicts the RFID reading - we may not be sure from these few values which sensor is most inaccurate, but perhaps by now you will trust that the gaussians expressing our beliefs will correctly handle these cases. It's a bit hard to talk about while working with 1D problems, so we will take this topic up in great detail in the next chapter where we develop multidimensional Kalman filters. Remark\n",
+      "\n",
+      "Besides that issue, we are modelling the noise in our sensors using gaussians which model their real world performance. We are multiplying the gaussians (probabilities) when we get a new position measurement, adding the gaussians when we get a movement update. This is algorithmically correct (this is how the histogram filter works) and mathematically correct - why wouldn't it work? \n",
+      "\n",
+      "#### Summary\n",
+      "This takes some time to assimulate. To truly understand this you will probably have to work through this chapter several times. I encourage you to change the various constants and observe the results. Convince yourself that gaussians are a good representation of a unimodal belief of something like the position of a dog in a hallway. Then convince yourself that multiplying gaussians truly does compute a new belief from your prior belief and the new measurement. Finally, convince yourself that if you are measuring movement, that adding the gaussians correctly updates your belief. That is all the Kalman filter does. Even now I alternate between complacency and amazement at the results. \n",
+      "\n",
+      "If you understand this, you will be able to understand multidimensional Kalman filters and the various extensions that have been make on them. If you do not fully understand this, I strongly suggest rereading this chapter until you do understand it. Try implementing the filter from scratch, just by looking at the equations and reading the text. Change the constants. Maybe try to implement a different tracking problem, like tracking stock prices. Experimentation will build your intuition and understanding of how these marvelous filters work.\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 31
     }
    ],
    "metadata": {}

Multidimensional Kalman Filters.ipynb

@@ -0,0 +1,285 @@
+{
+ "metadata": {
+  "name": ""
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "# Multidimensional Kalman Filters\n",
+      "\n",
+      "The techniques in the last chapter are very powerful, but they only work in one dimension. The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n",
+      "\n",
+      "\n",
+      "## Multivariate Normal Distributions\n",
+      "\n",
+      "What might a multivariate (meaning multidimensional) normal distribution look like? Our goal is to be able to represent a normal distribution across multiple dimensions. Consider the 2 dimensional case. Let's say we believe that x = 2 and y = 7. Therefore we can see that for N dimensions, we need N means, like so:\n",
+      "$$ \\mu = \\begin{bmatrix}{\\mu}_1\\\\{\\mu}_2\\\\ \\vdots \\\\{\\mu}_n\\end{bmatrix} \n",
+      "$$\n",
+      "\n",
+      "Therefore for this example we would have\n",
+      "$$\n",
+      "\\mu = \\begin{bmatrix}2\\\\7\\end{bmatrix} \n",
+      "$$\n",
+      "\n",
+      "The next step is representing our variances. At first blush we might think we would also need N variances for N dimensions. We might want to say the variance for x is 10 and the variance for y is 8. While this is possible, it does not consider the more general case. For example, suppose we were tracking house prices vs total $m^2$ of the floor plan. These numbers are *correlated*. It is not an exact correlation, but in general houses in the same neighborhood are more expensive if they have a larger floor plan. We want a way to express not only what we think the variance is in the price and the $m^2$, but also the degree to which they are correlated. It turns out that we use a matrix to denote this:\n",
+      "\n",
+      "$$\n",
+      "\\Sigma = \\begin{pmatrix}\n",
+      "  {\\sigma}_{1,1} & {\\sigma}_{1,2} & \\cdots & {\\sigma}_{1,n} \\\\\n",
+      "  {\\sigma}_{2,1} &{\\sigma}_{2,2} & \\cdots & {\\sigma}_{2,n} \\\\\n",
+      "  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n",
+      "  {\\sigma}_{n,1} & {\\sigma}_{n,2} & \\cdots & {\\sigma}_{n,n}\n",
+      " \\end{pmatrix}\n",
+      "$$\n",
+      "\n",
+      "This is called the covariance matrix, and is probably a bit confusing at the moment. Rather than explain the math in detail at the moment, we will take our usual tactic of building our intuition first with various physical models. \n",
+      "\n",
+      "So here is the full equation for the multivarate normal distribution.\n",
+      "\n",
+      "$$\\mathcal{N}(\\mu,\\,\\Sigma) = (2\\pi)^{-\\frac{n}{2}}|\\Sigma|^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(\\mathbf{x}-\\mu)'\\Sigma^{-1}(\\mathbf{x}-\\mu) }$$\n",
+      "\n",
+      "I urge you to not try to remember this function. We will program it once in a function and then call it when we need to compute a specific value. However, if you look at it briefly you will note that it looks quite similar to the univarate normal distribution except it uses matrices instead of scalar values. If you are reasonably well-versed in linear algebra this equation should look quite managable; if not, don't worry, the python is coming up next!\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy as np\n",
+      "import math\n",
+      "def multivariate_gaussian (x, mu, cov):\n",
+      "    n = len(x)\n",
+      "    det = np.sqrt(np.prod(np.diag(cov)))\n",
+      "    frac = (2*math.pi)**(-n/2.) * (1./det)\n",
+      "    fprime = (x - mu)**2\n",
+      "    return frac * np.exp(-0.5*np.dot(fprime, 1./np.diag(cov)))\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 57
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's use it to compute a few values just to make sure we know how to call and use the function, and then move on to more interesting things.\n",
+      "\n",
+      "First, let's find the probability for our dog being at 2.5, 7.3 if we believe he is at (2,7) with a variance of 4 for x and a variance of 5 for y. This function requires us to pass everything in as numpy arrays (we will soon provide a more robust version that works with numpy matrices, numpy arrays, and/or scalars in any combinations. That code contains a lot of boilerplate which obscures the algorithm).\n",
+      "\n",
+      "So we set x to (2.5,7.3)"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = np.array([2.5, 7.3])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 83
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Next, we set the mean of our belief:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "mu = np.array([2,7])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 84
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Finally, we have to define our covariance matrix. In the problem statement we did not mention any correlation between x and y, and we will assume there is none. This makes sense; a dog can choose to independently wander in either the x direction or y direction without affecting the other. If there is no correlation between the values you just fill in the diagonal of the covariance matrix with the variances:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "cov = np.array([[4,0],[0,5]])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 92
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now just call the function"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "print multivariate_gaussian (x,mu,cov)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "0.0341841496392\n"
+       ]
+      }
+     ],
+     "prompt_number": 93
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's check the probability for the dog being at exactly (2,7)"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = np.array([2,7])\n",
+      "print multivariate_gaussian (x,mu,cov)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "0.0355881271709\n"
+       ]
+      }
+     ],
+     "prompt_number": 94
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "These numbers are not easy to interpret. Let's plot this in 3D, with the z coordinate being the probability."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from mpl_toolkits.mplot3d import Axes3D\n",
+      "xs, ys = arange (-8, 13, .1), arange (-8, 20, .1)\n",
+      "xv, yv = meshgrid (xs, ys)\n",
+      "\n",
+      "zs = np.array([multivariate_gaussian(np.array([x,y]),mu,cov) for x,y in zip(np.ravel(xv), np.ravel(yv))])\n",
+      "zv = zs.reshape(xv.shape)\n",
+      "\n",
+      "ax = plt.figure().add_subplot(111, projection='3d')\n",
+      "ax.plot_surface (xv, yv, zv)\n",
+      "show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAV0AAADtCAYAAAAcNaZ2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3c50b10>"
+       ]
+      }
+     ],
+     "prompt_number": 104
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The result is clearly a 3D bell shaped curve. We can see that the gaussian is centered around (2,7), and that the probability quickly drops away in all directions.\n",
+      "\n",
+      "As beautiful as this is, it is perhaps a bit hard to get useful information. For example, it is not easy to tell if x and y both have the same variance or not. So for most of the rest of this book we will display multidimensional gaussian using contour plots.\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy.linalg as linalg\n",
+      "def plot_sigma_ellipse(x, y, cov):\n",
+      "    # Get the sigma ellipse by transforming a circle by the cholesky decomposition\n",
+      "\n",
+      "    L = linalg.cholesky(cov)\n",
+      "    t = linspace(0,2*pi,100)\n",
+      "    unit_circle = np.array([cos(t), sin(t)])\n",
+      "\n",
+      "    E1 = L.dot(unit_circle)\n",
+      "    axis('equal')\n",
+      "    plot(E1[0]+x, E1[1]+y)\n",
+      "    scatter(x,y,marker='+') # mark the center\n",
+      "    show()\n",
+      "    \n",
+      "cov = array ([[2,0],[0,2]])\n",
+      "plot_sigma_ellipse (2,7,cov)\n",
+      "\n",
+      "cov = array ([[2,0],[0,12]])\n",
+      "plot_sigma_ellipse (2,7,cov)\n",
+      "\n",
+      "cov = array ([[2,3],[1,2.5]])\n",
+      "plot_sigma_ellipse (2,7,cov)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAEACAYAAACuzv3DAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x3641d50>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x365dfd0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x57e7410>"
+       ]
+      }
+     ],
+     "prompt_number": 155
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}

gaussian.py

@@ -70,7 +70,6 @@ def norm_plot (mean, var):
     xs = np.arange (min_x, max_x, 0.1)
     ys = [gaussian (x,23,5) for x in xs]
     plt.plot (xs,ys)
-    plt.show()
 
 if __name__ == '__main__':
     from scipy.stats import norm
@@ -88,4 +87,3 @@ if __name__ == '__main__':
     assert rv.pdf(1.2) == x2
     assert abs(x2- x3) < 0.00000001
     print "all tests passed"
-